This paper introduces a novel concept of Boolean function{based hypergraph with respect to any given T. B. T(total binary truth table). This study de, nes a notation of kernel set on switching functions and proves that every T. B. T corresponds to a Minimum Boolean expression via kernel set and presents some conditions on T. B. T to obtain a Minimum irreducible Boolean expression from switching functions. Finally, we present an algorithm and so Python programming(with complete and original codes) such that for any given T. B. T, introduces a Minimum irreducible switching expression.

Comer introduced a class of hypergroups, using the name of polygroups. He emphasized the importance of polygroups, by analyzing them in connections to graphs, relations, Boolean and cylindric algebras. Indeed, polygroups are multi valued systems that satisfy group like axioms. Given a polygroup with a finite generating set, we can form a Cayley hypergraph for that polygroup with respect to that generating set. This helps us to better understand and investigate polygroup structures. More precisely,in this paper, we introduce the construction of Cayley hypergraphs over polygroups, say $CH(\mathbf{P},S)$ such that $\mathbf{P}$ is a polygroup and $\langle S\rangle =P$. We investigate some properties of them. It is well known to give a constructing for building a big polygroup from two small ones. This structure is called extensionof polygroups. In particular, we describe the connection between Cayley hypergraphs over extension of two polygroups and Cartesian product of two Cayley hypergraphs.

This paper presents a new method for simplification of Boolean functions based on Boolean differences. The proposed method is applicable to various forms of Boolean functions, including truth tables and Binary Decision Diagrams (BDDs). The Boolean differences are extended to cover the truth tables with don't-care components and cutset graphs in BDDs. The results of simplification agree with Quine-McCluskey and ESPRESSO methods. Experimental tests on MCNC and Berkeley PLA benchmarks show that the proposed method gains a performance of 1.5-10 times faster than ESPRESSO. The algorithms of the proposed method are implemented in Java/Perl/C++, and a toolset for logic function simplification is developed.

For an associated combinatorial structure with Lie algebra $\mathbf{g}_n$ of upper triangular matrices, an allowable, forbidden, and the graphs that are not associated with $\mathbf{g}_n$ of any three vertices are determined. This work also introduces a neoteric association of hypergraph with Lie algebra of upper triangular matrix $\mathcal{G}_n$ for an element of Lie algebra $\mathbf{g}_n$. The properties of this structure are analyzed, characterized and have been presented as an algorithm for finite order.